A281976 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z <= w such that both x and x + 24*y are squares.

1, 2, 3, 2, 2, 3, 3, 2, 1, 3, 4, 2, 1, 2, 2, 2, 2, 3, 5, 2, 3, 3, 2, 1, 1, 4, 5, 4, 2, 2, 4, 3, 3, 3, 6, 2, 6, 5, 3, 3, 3, 7, 6, 2, 2, 5, 4, 1, 2, 3, 7, 6, 8, 4, 5, 5, 2, 4, 5, 2, 3, 5, 3, 4, 2, 5, 9, 4, 5, 4, 5, 1, 3, 5, 4, 5, 5, 4, 2, 3, 3

Offset

0,2

Comments

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 16^k*m (k = 0,1,2,... and m = 8, 12, 23, 24, 47, 71, 168, 344, 632, 1724).

By the linked JNT paper, any nonnegative integer can be written as the sum of a fourth power and three squares.

We have verified a(n) > 0 for all n = 0..10^7.

See also A281977, A282013 and A282014 for similar conjectures.

a(n) <= A273404(n). Starts to differ from A273404 at n=145. - R. J. Mathar, Feb 12 2017

Qing-Hu Hou at Tianjin Univ. has verified a(n) > 0 for all n = 0..10^10.

I would like to offer 2400 US dollars for the first proof of my conjecture that a(n) > 0 for any nonnegative integer n. - Zhi-Wei Sun, Feb 14 2017

Links

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.

Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Zhi-Wei Sun, The 24-conjecture with $2400 prize, a message to Number Theory List, Feb. 14, 2017.

Example

a(8) = 1 since 8 = 0^2 + 0^2 + 2^2 + 2^2 with 0 = 0^2 and 0 + 24*0 = 0^2.
a(12) = 1 since 12 = 1^2 + 1^2 + 1^2 + 3^2 with 1 = 1^2 and 1 + 24*1 = 5^2.
a(23) = 1 since 23 = 1^2 + 2^2 + 3^2 + 3^2 with 1 = 1^2 and 1 + 24*2 = 7^2.
a(24) = 1 since 24 = 4^2 + 0^2 + 2^2 + 2^2 with 4 = 2^2 and 4 + 24*0 = 2^2.
a(47) = 1 since 47 = 1^2 + 1^2 + 3^2 + 6^2 with 1 = 1^2 and 1 + 24*1 = 5^2.
a(71) = 1 since 71 = 1^2 + 5^2 + 3^2 + 6^2 with 1 = 1^2 and 1 + 24*5 = 11^2.
a(168) = 1 since 168 = 4^2 + 4^2 + 6^2 + 10^2 with 4 = 2^2 and 4 + 24*4 = 10^2.
a(344) = 1 since 344 = 4^2 + 0^2 + 2^2 + 18^2 with 4 = 2^2 and 4 + 24*0 = 2^2.
a(632) = 1 since 632 = 0^2 + 6^2 + 14^2 + 20^2 with 0 = 0^2 and 0 + 24*6 = 12^2.
a(1724) = 1 since 1724 = 25^2 + 1^2 + 3^2 + 33^2 with 25 = 5^2 and 25 + 24*1 = 7^2.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0; Do[If[SQ[n-x^4-y^2-z^2]&&SQ[x^2+24y], r=r+1], {x, 0, n^(1/4)}, {y, 0, Sqrt[n-x^4]}, {z, 0, Sqrt[(n-x^4-y^2)/2]}]; Print[n, " ", r]; Continue, {n, 0, 80}]

Crossrefs

Cf. A000118, A000290, A000583, A270969, A273404, A281939, A281941, A281975, A281977, A281980, A282013, A282014.

Sequence in context: A317420 A256795 A273404 * A300708 A240755 A309806

Adjacent sequences: A281973 A281974 A281975 * A281977 A281978 A281979

Keyword

nonn

Author

Zhi-Wei Sun, Feb 04 2017

Status

approved